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Determinant and Weyl anomaly of Dirac operator:

a holographic derivation

R Aros and D E Dıaz

Universidad Andres Bello, Facultad de Ciencias Exactas, Departamento deCiencias Fisicas, Republica 220, Santiago, Chile

E-mail: raros,[email protected]

Abstract. We present a holographic formula relating functional determinants:the fermion determinant in the one-loop effective action of bulk spinors in anasymptotically locally AdS background, and the determinant of the two-pointfunction of the dual operator at the conformal boundary. The formula originatesfrom AdS/CFT heuristics that map a quantum contribution in the bulk partitionfunction to a subleading large-N contribution in the boundary partition function.We use this holographic picture to address questions in spectral theory andconformal geometry. As an instance, we compute the type-A Weyl anomaly andthe determinant of the iterated Dirac operator on round spheres, express the latterin terms of Barnes’ multiple gamma function and gain insight into a conjectureby Bar and Schopka.

1. Introduction

Ever since its appearance almost fifteen years ago in the form of Maldacena’sconjecture, the AdS/CFT correspondence [1, 2, 3] has been a successful tool to addressquestions concerning strongly-coupled systems. Many developments depart from theoriginal canonical formulation in pure anti-de Sitter (AdS) spacetime (mostly stillrestricted to classical (super)gravity in the bulk); for instance, finite-temperatureeffects on the boundary theory led to consideration of AdS black holes as bulkbackground geometries. This novel holographic approach, phenomenological in nature,covers an increasing amount of physical situations ranging from strongly coupledquark-gluon plasma and condensed matter systems to cosmological singularities andblack hole physics. At present, this ambitious but conjectural program seems tosucceed at a qualitative level (cf. [4]).

In contrast, quantitative and exact results in AdS/CFT correspondence are tobe found in the interplay with mathematics; in for instance conformal geometryand spectral theory. Geometric roots of AdS/CFT date back to the seminal workof Fefferman and Graham [5, 6] that addresses conformal geometry on a compactmanifold as geometry at the conformal infinity of space-filling Poincare metrics.Conversely, AdS/CFT revealed interesting conformal invariants, e.g. Q-curvature,that arise in the volume renormalization of these Poincare metrics and triggered newdevelopments in conformal geometry (cf. [7, 8]).

The present contribution will focus precisely on these latter aspects of theduality, where the foreseeable progress seems modest but solid. We deal with‘holographic formulas’ as special entries in the AdS/CFT dictionary, relating one-loop

http://arxiv.org/abs/1111.1463v2

Spinor holographic formula 2

determinants for bulk fields in asymptotically AdS backgrounds and determinants ofcorrelation functions of the dual operators at the boundary. They originate in quantumrefinements of the duality where one-loop corrections in the gravity side are mappedto sub-leading terms in the large-N expansion of the boundary theory. Interestingeffects in for instance thermodynamics and transport phenomena on the boundaryare captured by the holographic correspondence only after inclusion of quantum one-loop effects in the bulk (cf. [9, 10]). A systematic study of bulk scalars has led toa holographic formula which has been verified in certain cases amenable to analyticevaluation; these bulk geometries include pure and thermal AdS, the BTZ black hole,and other quotients or orbifolds of AdS [11, 12, 13, 14].

Our aim now is to show that also for bulk spinors an analogous holographicformula can be established. Explicit computations are performed for the ball model ofhyperbolic space which embrace several scattered results in the literature. In this casethe bulk side and the role of Barnes’ gamma function, already explored in [15, 16],can be further exploited to get a closed formula for the determinant of the Diracoperator on round spheres, an interesting result that seems to have escaped notice. Auniversal formula for the type-A trace anomaly of Dirac operator is as well obtainedin this holographic way. These explicit results contain previous ones found in relationwith proposals for a c-theorem in dimensions other than two; they include Cardy’s a-theorem [17, 18], universal terms in entanglement entropy [19, 20] and F-theorem [21].

We start in section 2 with a review of bulk spinors in AdS/CFT and its doublequantization to predict an O(1) quantum contribution to the partition functions. Nextwe analyze the dual picture at the boundary in section 3 in order to detect this O(1)contribution to the partition function on the CFT side. In section 4 we write down thespinor holographic formula. In section 5, the pure AdS bulk geometry is consideredas an instance where both sides of the formula can be worked out in detail. Section6 is concerned with the Dirac operator at the boundary and the application of theholographic formula to read off the universal part in the associated Polyakov formulas,the type-A trace anomaly as well as the functional determinant on round spheres. Insection 7 we examine several scattered results which are now encompassed by ourcalculations. Concluding remarks are given in section 8.

2. Bulk spinors and double quantization

The role of bulk spinors in AdS/CFT has, of course, been extensively studied sincethe early days of the correspondence; a non-exhaustive list includes [22, 23, 24, 25].To begin with, we choose the bulk side and review the features relevant to our presentconcern, namely, the spinor version of the holographic formula.

Consider a bulk metric that approaches asymptotically the Poincare half-spacemodel for the Euclidean section of AdSn+1, that is, hyperbolic space Hn+1

ds2 =dz2 + d~x2

z2. (1)

The solutions of Dirac equation (/∇ + m)ψ = 0, with positive mass m > 0 fordefiniteness, behave near the conformal boundary z = 0 as‡

ψ ∼ zλ−ψo(~x) + zλ+χo(~x) , (2)

‡ Strictly speaking, valid for 0 < m < 12.


with λ± = n2 ± m, and the boundary data ψo and χo belong to the eigenspace of

the flat Dirac gamma associated to the z−direction, Γo, with eigenvalues −1 and +1,respectively.

The requirement of regularity at the deep interior, z → ∞, imposes a linearrelation between ψo and χo given by convolution with the scattering operator χo =S(λ) ∗ ψ0, or equivalently, with the kernel associated to the two-point function of thedual operator at the boundary. It is this on-shell relation which ultimately leads,upon functional differentiation of the action with respect to the boundary source, tothe corresponding two-point correlator

〈O+ O+〉 ∼~Γ · (~x− ~y)

|~x− ~y|1+n+2m. (3)

The standard AdS/CFT recipe contemplates ψo as the source and χo asexpectation value of the dual primary operator with conformal dimension λ+ atthe boundary. In the variational approach to the path integral, the sum overhistories is preformed with ψo and ψo prescribed (q’s), whereas χo and χo are freeto vary (canonically conjugate momenta p’s). Nonetheless, a crucial observation(cf. [18, 21, 26]) is that whenever 0 < m < 1/2 the dimension λ− is above theunitarity bound, in this case one is free to interchange the roles of ψo and χo to obtainthe two-point function of a dual operator of dimension λ−

〈O− O−〉 ∼~Γ · (~x− ~y)

|~x− ~y|1+n−2m. (4)

2.1. O(1) contribution to the partition function

That is very much the picture at the classical level in the bulk. Considering nowquantum fluctuations around the on-shell configuration, there are therefore twopossible quantizations of the bulk spinor: the conventional one with ψo set to zeroand another, alternative one, with χo set to zero. Whenever the mass of the bulkspinor lies in the window 0 < m < 1

2 , both kinds of modes are normalizable. Theircontribution to the partition function can be computed in the standard way via theGreen function for the conventional modes (λ+), and analytically continue to accountfor the alternative modes (λ−). With this choice at hand, we can emulate the scalarcase since now double quantization for bulk spinors is established. The relative changein the partition function, upon functional integration of the quantum fluctuations atquadratic order, is then given by the ratio of the associated functional determinants:

Z+grav

Z−grav

=det+{/∇X

+m}

det−{/∇X+m}

. (5)

3. Boundary double-trace deformation

These two choices of asymptotic behavior correspond in AdS/CFT to two CFT’s whichshare the same field content but differ in the dimension of the fermionic operator O,dual to the bulk spinor. The UV CFT with O−, perturbed by the relevant double-tracedeformation O2

−, flows into the IR CFT with O+ (cf. [18, 21, 26]).


3.1. O(1) contribution to the partition function: a shortcut

To get a handle on the relative change in the CFT partition functions at the end pointsof the RG flow, instead of considering the auxiliary field trick as in [18, 21], we simplyadapt an argument given in [27] to relate ZUV to ZIR. Namely, in the path integralof the UV CFT we promote the sources η and η to dynamical fields and integrate overthem

ZIR

ZUV=

∫

DηDη 〈 exp

∫

(ηO− + O−η) 〉 . (6)

The expectation value can be approximated, at leading large N due to the factorizationof the correlation functions, by

exp

∫

η〈O− O−〉η , (7)

and the Gaussian integral results in the functional determinant of the two-pointfunction

ZIR

ZUV= det 〈O− O−〉 , (8)

or, alternatively,

ZUV

ZIR= det 〈O+ O+〉 . (9)

4. The holographic formula

We have now all necessary ingredients to write down the ‘spinor holographic formula’that stems from the postulated equality of the partition functions in AdS/CFTcorrespondence, at subleading O(1):

det−{/∇X+m}

det+{/∇X+m}

= det 〈Oλ Oλ〉M (10)

The ‘+’ means that we compute with the standard λ in the asymptotic behavior ofthe bulk spinor, whereas ‘−’ means the analytic continuation λ→ n−λ. As usual, tomake sense out of this formula one needs to tame the divergencies that arise due to theIR divergent volume of AdS and the UV divergent short distance singularities. Bothdivergencies turn out to be tied by the so-called IR/UV in AdS/CFT correspondence.

This formula conjecturally applies to bulk geometries X which are Euclideansections of asymptotically locally AdS (ALAdS). In particular, when the conformalinfinity M belongs to the conformal class of the standard round spheres, and thereforeconformally flat, the bulk is locally AdS and the IR-divergent volume of AdS factorizes.One can then read off an O(1) contribution to the holographic trace anomaly in evenn, just as in the case of a bulk scalar [3, 7, 28]. Alternatively, from the differenceof one-loop effective actions one can compute the holographic type-A trace anomalycoefficient a following the general recipe of [29]. The behavior of this coefficient foreven n and of a related quantity for odd n, in the cases we will explore, gives support toa conjectured c-theorem valid in all dimensions [20] § and to a F-theorem proposal [21]as well.

§ This promises to answer the disparity pointed out in [30] that, although the computation on theAdS side contemplates even and odd dimensions on equal footing, it is not clear how to translate the“holographic” central charge into field theory language in the case of odd-dimensional CFT’s.


5. The canonical case

As might be expected, the ball model for hyperbolic space (Euclidean AdS) turns outto be the simplest bulk background where calculations can be spelled out in detailand related to the CFT on the conformal boundary (the conformally flat class of thestandard round sphere).

5.1. Bulk

The effective action for a Dirac spinor in hyperbolic space has been recentlyrevisited [15, 16] in connection with a curious gauge-gravity duality where Barnes’multiple gamma function plays a central role. We briefly survey the relevant steps inthe computation of the effective action

S+grav = − log det{/∇+m} . (11)

In terms of the Green’s function, one has (/∇+m)D = −I,

S+grav =

∫ m

trD(n+1) , (12)

where also the spinor indices are traced out. We refer to [15, 16] for details ofthe implementation of dimensional regularization and the nontrivial role of the bulkvolume. In all, one gets the remarkable result, valid for both even and odd dimensions,in terms of Barnes’ multiple gamma

logdet+{/∇+m}

det−{/∇+m}= −21+⌊n

2 ⌋ · logΓn+1(

n+12 +m)

Γn+1(n+12 −m)

. (13)

In addition, whenever n is even one can read off the trace anomaly as in thescalar case [11, 12]. In the present case one essentially gets the integral of the spinorPlancherel‖ measure(cf. [31]) times the volume anomaly Ln+1 = 2(−π)

n2 /Γ(1 + n

2 )[

2

(2π)n2

∫ m

0

dµ(12 + µ)n

2· (12 − µ)n

2

(12 )n2

]

· Ln+1 . (14)

5.2. Boundary

For the round n-sphere as conformal boundary, the knowledge of the eigenvaluesof the two-point function 〈Oλ Oλ〉Sn and their degeneracies allows a brute-forcecomputation of the corresponding functional determinant. The appropriate basis isthat of spinor spherical harmonics, and the eigenvalues¶ and degeneracies have beenrecently computed in connection with fermionic double-trace deformations [18]

eigenvalues: ±Γ(l + n/2 + ν + 1/2)

Γ(l + n/2− ν + 1/2), (15)

degeneracies: 2⌊n2 ⌋ (l + n− 1)!

l! (n− 1)!. (16)

‖ Difference of the effective Lagrangians.¶ The eigenvalues can be also read off form the scattering problem in Hn+1 [32].


The formal trace is then assembled+ as follows:

log det 〈Oλ Oλ〉Sn = 21+⌊n2 ⌋

∞∑

l=0

(n)ll!

logΓ(l + n+1

2 + ν)

Γ(l + n+12 − ν)

, (17)

and it can be worked out within dimensional regularization as in [11, 21]

log det 〈Oλ Oλ〉Sn = −21+⌊n2 ⌋ Γ(−n)

∫ ν

0

dµ

{

Γ(n+12 + µ)

Γ(1−n2 + µ)

+ (µ → −µ)

}

. (18)

This very same regularized answer we have already found in [16] (eqn.5) whereBarnes’s multiple gamma turned up. The renormalized value can be written, modulopolynomial and logarithmic terms (we refer again to [16] for further details and eqn.6),as the following quotient of Barnes’ multiple gamma functions

log det 〈Oλ Oλ〉Sn = 21+⌊n2 ⌋ log

Γn+1(n+12 + ν)

Γn+1(n+12 − ν)

. (19)

As explained in [16], the gamma factor in front of (18) is deceiving: only for n eventhere is a pole and from the residue one can read off the infinitesimal (integrated)conformal anomaly∗ under conformal rescaling of the metric

22+⌊n2 ⌋ (−1)n/2

n!

∫ ν

0

dµ (1

2+ µ)n

2· (1

2− µ)n

2. (20)

At this point we have a perfect match between bulk and boundary computations. Toillustrate the usefulness of this result, besides being a explicit corroboration of theholographic formula, we study a particular value of the spinor mass that unveils theDirac operator on the boundary and connects with a vast mathematical literature ondeterminants of differential operators on spheres (cf. [33]-[39]).

6. Holographic life of Dirac operator

There are two direct ways to identify the Dirac operator at the boundary.

• First, consider the action of the Dirac operator on the flat space two-pointfunction(eqn. 3)

/∇〈O+(~x) O+(~0)〉Rn ∼I

|~x|n+1+m. (21)

Here one can recognize the Laplacian ∇2 in the limit m → 12 , in a distributional

sense.

• Second, by simple inspection of the eigenvalues of the kernel 〈O+ O+〉Sn on then-sphere (eqn.15) in the same limit m→ 1

2

± (n

2+ l) . (22)

+ In dimensional regularization, the sum over degeneracies of a constant term vanishes, this is whywe do not worry much about the fact that half of the eigenvalues are negative.∗ The shorthand notation involves Pochhammer’s symbol (x)n ≡

Γ(x+n)Γ(x)

.


6.1. Type A trace anomaly and Polyakov formulas

We are interested in the universal type A component of the trace anomaly. Forconformally related metrics g = e2wg, in the conformal flat class, Branson conjecturedthe following Polyakov formula

− logdet /∇

2

det /∇2 = c(n)/∇2

∫

M

w(

Qn dvg + Qn dvg

)

+ ... (23)

This very same structure we read from the bulk computation, the Q-curvature termscome from the finite conformal variation of the renomalized volume [40, 41, 42] andthe overall coefficient c(n)

/∇2is obtained from the effective Lagrangian at m = 1

2

c(n)/∇2

= 4cn

2

(2π)n2

∫ 12

0

dν(12 + ν)n

2· (12 − ν)n

2

(12 )n2

, (24)

where ck = (−1)k

22kk!(k−1)!. All values reported in [33] for c(n)

/∇2are correctly reproduced by

this formula.

6.2. Determinant of iterated Dirac on spheres

The particular value of the determinant of the scattering operator at the mass value12 results in the following remarkable expression in terms of Barnes’ multiple gammafunction, after use of recurrence (A.3)

− log det /∇2 = 4 · 2[n2

]

· log Γn (n

2) . (25)

This compact expression seems to correctly reproduce all zeta-regularized valuesreported in the literature(cf. [35, 39]).

A small digression on a conjecture put forward by Bar and Shopka [35]: theyobserved that the numerical values of these determinants tend to 1 as the dimension ngrows; this was proved in [37] not only for the Dirac operator, but also for the Yamabeor conformal Laplacian on the n-sphere. Our result indicates that the above findingsamount to establishing the limiting value of Barnes’ gamma Γn(n/2) as n → ∞.Furthermore, we can interpret quite naturally this limiting value by looking at thebulk side of the holographic formula: the two boundary conditions coincide as ngrows, or alternatively, λ± → n

2 so that both determinants in the quotient approachone another. This very same argument would predict the same limiting value for allother operators on the right side of similar holographic formulas; this is the case for thedeterminants of GJMS operators on round spheres [12]. In consequence, we predictthe same limiting value of unity; a prediction that can be probed by the asymptoticanalysis of the explicit results obtained in [38].

7. C-theorem proposals and entanglement entropy

Our computations contain and, at times, generalize several scattered results that hadbeen independently derived, most of them in the pursuit of extensions of the C-theoremto higher dimensions and in connection with certain universal terms in entanglemententropy. We briefly list few of them.


Holographic C-charge at O(1):

• n = even, relative change at order O(1) in Cardy’s central charge computedin [18]. CUV − CIR > 0 in the mass window 0 < m ≤ 1

2 . Agreement with theuniversal log-term in entanglement entropy [20].

We find agreement between bulk and boundary outcomes - tables (1) and (2)in [18]- and our eqns. (14) and (20), respectively. Our calculation accounts forthe overall coefficient as well.

• n = odd, relative change at order O(1) in one-loop effective action, candidate forcentral charge, computed in [18]. CUV −CIR > 0 in the mass window 0 < m ≤ 1

2 .Agreement with the universal constant term in entanglement entropy [20].

We find agreement between bulk outcome -integral of eqn.(3.37) in [18]- and oureqn.(13). We are also able to fill in the ‘hole’ left in [18] by identifying this finitecontribution on the CFT side, including the overall coefficient as well (eqn.19).

F-coefficient of odd-dimensional CFT’s:

• F-coefficients for free massless Dirac fermions on odd-spheres, table (2) in [21].

This agrees with our result♯ in eqn.(25) and, of course, with the values reportedin [35, 37]. The decrease of the numerical values in the above table is preciselythe hint for the conjecture by Bar and Schopka.

• Under RG flow triggered by a fermionic double-trace deformation, at leadinglarge N, the change in free energy in three dimensions is given by eqn.(82),[21].FUV − FIR > 0 in the mass window 0 < m ≤ 1

2 .

This coincides again with our bulk(eqn.13) and boundary(eqn.19) results.

Entanglement entropy:

• Universal terms in entanglement entropy [20]: logarithmic and constant in evenand odd dimensions, respectively. In the case of a free boson, they coincide withthe holographic anomaly and determinant of Yamabe or conformal Laplacian op-erator, for even [38] and odd [43] dimensions, respectively.

This matching holds as well for the entanglement entropy of a free Dirac spinorand the trace anomaly and determinant of the Dirac operator on n-spheres; nowunder the guise of holographic C-charge at O(1).

8. Conclusion

In all, we have worked out the spinor version of the holographic formula which connectsfunctional determinants of bulk spinor with that of the two point function of the dualoperator at the boundary. The case of pure AdS allows for explicit computations

♯ The evaluation of the corresponding Barnes’ gammas can be performed with the relations collectedin Appendix A, and one can rewrite in terms of Riemann zeta by use of the relation ζ′(−2n) =(−1)n(2n)!

21+2nπ2n ζ(1 + 2n) .


which, in turn, encompass several results independently derived in other contexts. Inparticular, contact is made in the case of the Dirac operator on the boundary. Here wehave obtained a compact expression for the determinant on spheres in terms of Barnes’gamma function as well as the generic type-A trace anomaly in any even dimension.We have also gained insight into the conjecture by Bar and Schopka and unveiled itspossible holographic roots.

Acknowledgments

This work was partially funded through fondecyt-chile 11110430 and UNAB DI-21-11/R.

Appendix A. Barnes’ multiple gamma: disambiguation

There are several choices for the normalization of Barnes’ multiple gamma functionΓn(z). For definiteness, we stick to the convention in [39, 44, 45] and list few propertiesthat are relevant to our calculations.

• Its logarithm can be written in terms of derivatives of Hurwitz zeta function

log Γn(z) =

n−1∑

k=0

bn,k(z) · ζ′(−k, z) , (A.1)

where bn,k(z) is a polynomial in z with Stirling numbers of the first kind s(n, j)in its coefficients

bn,k(z) =(−1)n−1−k

(n− 1)!

n−1∑

j=k

(jk

) · s(n, j + 1) · zj−k . (A.2)

• Recurrence or ladder relation:

Γn+1(1 + z) =Γn+1(z)

Γn(z). (A.3)

• Pascal triangle by successive applications of the ladder relation:

log Γn(m+ z) =m∑

l=0

(−1)l (ml

) · log Γn−l(z) , 0 ≤ m ≤ n− 1 . (A.4)

• Particular values in terms of derivatives of Riemann zeta function††:

log Γn(1) =n−1∑

k=0

bn,k(1) · ζ′(−k) , (A.5)

log Γn(1

2) =

n−1∑

k=0

bn,k(1

2) · (2−k − 1) · ζ′(−k)− log 2

n−1∑

k=0

bn,k(1

2)2−kBk+1

k + 1, (A.6)

where Bn are Bernoulli numbers.

††Relation A.5 can be further simplified to log Γn(1) = 1(n−1)!

∑n−1j=1 |s(j, n − 1)| · ζ′(−k), cf. [34]

where their Pn is just the inverse of the gamma we are using.


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[35] C. Bar and S. Schopka, The Dirac determinant of spherical space forms, Geo. Anal. andNonlinear PDEs, Springer, Berlin (2003) 39–67.

[36] J. S. Dowker, K. Kirsten, The Barnes zeta function, sphere determinants andGlaisher-Kinkelin-Bendersky constants, Anal. Appl.3 (2005) 45-68, [hep-th/0301143].

[37] N. M. Møller, Dimensional asymptotics of determinants on Sn, and proof of Bar-Schopkaconjecture, Math. Ann. 343 (2009) 35–51.

[38] J. S. Dowker, Determinants and conformal anomalies of GJMS operators on spheres, J. Phys.A A44 (2011) 115402, [arXiv:1010.0566 [hep-th]].

[39] Y. Yamasaki, Factorization formulas for higher depth determinants of the Laplacian on then-sphere, [arXiv:1011.3095[math.NT]].

[40] A. Chang, J. Qing and P. Yang, On the renormalized volumes for conformally compactEinstein manifolds, [arXiv:math.DG/0512376].

[41] S. Carlip, Dynamics of asymptotic diffeomorphisms in (2+1)-dimensional gravity, Class.Quant. Grav.22 (2005) 3055-3060, [gr-qc/0501033].

[42] R. Aros, M. Romo and N. Zamorano, Conformal Gravity from AdS/CFT mechanism, Phys.Rev.D75 (2007) 067501, [hep-th/0612028].

[43] J. S. Dowker, Entanglement entropy for odd spheres, [arXiv:1012.1548 [hep-th]].[44] S. N. M. Ruijsenaars, On Barnes’ multiple zeta and gamma functions, Advances in

Mathematics 156 (2000) 107–132.[45] E. Friedman and S. N. M. Ruijsenaars, Shintani-Barnes zeta and gamma functions, Advances

in Mathematics 187 (2004) 362–395.

arX

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1463

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Determinant and Weyl anomaly of Dirac operator:

a holographic derivation

R Aros and D E Dıaz

Universidad Andres Bello, Facultad de Ciencias Exactas, Departamento deCiencias Fisicas, Republica 220, Santiago, Chile

E-mail: raros,[email protected]

Abstract. We present a holographic formula relating functional determinants:the fermion determinant in the one-loop effective action of bulk spinors in anasymptotically locally AdS background, and the determinant of the two-pointfunction of the dual operator at the conformal boundary. The formula originatesfrom AdS/CFT heuristics that map a quantum contribution in the bulk partitionfunction to a subleading large-N contribution in the boundary partition function.We use this holographic picture to address questions in spectral theory andconformal geometry. As an instance, we compute the type-A Weyl anomaly andthe determinant of the iterated Dirac operator on round spheres, express the latterin terms of Barnes’ multiple gamma function and gain insight into a conjectureby Bar and Schopka.

1. Introduction

Ever since its appearance almost fifteen years ago in the form of Maldacena’sconjecture, the AdS/CFT correspondence [1, 2, 3] has been a successful tool to addressquestions concerning strongly-coupled systems. Many developments depart from theoriginal canonical formulation in pure anti-de Sitter (AdS) spacetime (mostly stillrestricted to classical (super)gravity in the bulk); for instance, finite-temperatureeffects on the boundary theory led to consideration of AdS black holes as bulkbackground geometries. This novel holographic approach, phenomenological in nature,covers an increasing amount of physical situations ranging from strongly coupledquark-gluon plasma and condensed matter systems to cosmological singularities andblack hole physics. At present, this ambitious but conjectural program seems tosucceed at a qualitative level (cf. [4]).

In contrast, quantitative and exact results in AdS/CFT correspondence are tobe found in the interplay with mathematics; in for instance conformal geometryand spectral theory. Geometric roots of AdS/CFT date back to the seminal workof Fefferman and Graham [5, 6] that addresses conformal geometry on a compactmanifold as geometry at the conformal infinity of space-filling Poincare metrics.Conversely, AdS/CFT revealed interesting conformal invariants, e.g. Q-curvature,that arise in the volume renormalization of these Poincare metrics and triggered newdevelopments in conformal geometry (cf. [7, 8]).

The present contribution will focus precisely on these latter aspects of theduality, where the foreseeable progress seems modest but solid. We deal with‘holographic formulas’ as special entries in the AdS/CFT dictionary, relating one-loop

http://arxiv.org/abs/1111.1463v2


determinants for bulk fields in asymptotically AdS backgrounds and determinants ofcorrelation functions of the dual operators at the boundary. They originate in quantumrefinements of the duality where one-loop corrections in the gravity side are mappedto sub-leading terms in the large-N expansion of the boundary theory. Interestingeffects in for instance thermodynamics and transport phenomena on the boundaryare captured by the holographic correspondence only after inclusion of quantum one-loop effects in the bulk (cf. [9, 10]). A systematic study of bulk scalars has led toa holographic formula which has been verified in certain cases amenable to analyticevaluation; these bulk geometries include pure and thermal AdS, the BTZ black hole,and other quotients or orbifolds of AdS [11, 12, 13, 14].

Our aim now is to show that also for bulk spinors an analogous holographicformula can be established. Explicit computations are performed for the ball model ofhyperbolic space which embrace several scattered results in the literature. In this casethe bulk side and the role of Barnes’ gamma function, already explored in [15, 16], canbe further exploited to get a closed formula for the determinant of the Dirac operatoron round spheres, an interesting result that seems to have escaped notice. A universalformula for the type-A trace anomaly [17] of Dirac operator is as well obtained inthis holographic way. These explicit results contain previous ones found in relationwith proposals for a c-theorem in dimensions other than two; they include Cardy’s a-theorem [18, 19], universal terms in entanglement entropy [20, 21] and F-theorem [22].

We start in section 2 with a review of bulk spinors in AdS/CFT and its doublequantization to predict an O(1) quantum contribution to the partition functions. Nextwe analyze the dual picture at the boundary in section 3 in order to detect this O(1)contribution to the partition function on the CFT side. In section 4 we write down thespinor holographic formula. In section 5, the pure AdS bulk geometry is consideredas an instance where both sides of the formula can be worked out in detail. Section6 is concerned with the Dirac operator at the boundary and the application of theholographic formula to read off the universal part in the associated Polyakov formulas,the type-A trace anomaly as well as the functional determinant on round spheres.In section 7 we examine several scattered results closely related to our calculations.Concluding remarks are given in section 8, and conventions and useful identities forBarnes’ gammas are collected in an appendix.

2. Bulk spinors and double quantization

The role of bulk spinors in AdS/CFT has, of course, been extensively studied sincethe early days of the correspondence; a non-exhaustive list includes [23, 24, 25, 26].To begin with, we choose the bulk side and review the features relevant to our presentconcern, namely, the spinor version of the holographic formula.

Consider a bulk metric that approaches asymptotically the Poincare half-spacemodel for the Euclidean section of AdSn+1, that is, hyperbolic space Hn+1

ds2 =dz2 + d~x2

z2. (1)

The solutions of Dirac equation (/∇ + m)ψ = 0, with positive mass m > 0 fordefiniteness, behave near the conformal boundary z = 0 as‡

ψ ∼ zλ−ψo(~x) + zλ+χo(~x) , (2)

‡ Strictly speaking, valid for 0 < m < 12.


with λ± = n2 ± m, and the boundary data ψo and χo belong to the eigenspace of

the flat Dirac gamma associated to the z−direction, Γo, with eigenvalues −1 and +1,respectively.

The requirement of regularity at the deep interior, z → ∞, imposes a linearrelation between ψo and χo given by convolution with the scattering operator χo =S(λ) ∗ ψ0, or equivalently, with the kernel associated to the two-point function of thedual operator at the boundary. It is this on-shell relation which ultimately leads,upon functional differentiation of the action with respect to the boundary source, tothe corresponding two-point correlator

〈O+ O+〉 ∼~Γ · (~x− ~y)

|~x− ~y|1+n+2m. (3)

The standard AdS/CFT recipe contemplates ψo as the source and χo asexpectation value of the dual primary operator with conformal dimension λ+ at theboundary. In the variational approach to the path integral, the sum over histories ispreformed with ψo and ψo prescribed, whereas χo and χo are free to vary. Nonetheless,a crucial observation (cf. [19, 22, 27]) is that whenever 0 < m < 1/2 the dimensionλ− is above the unitarity bound, in this case one is free to interchange the roles of ψo

and χo to obtain the two-point function of a dual operator of dimension λ−

〈O− O−〉 ∼~Γ · (~x− ~y)

|~x− ~y|1+n−2m. (4)

2.1. O(1) contribution to the partition function

That is very much the picture at the classical level in the bulk. Consideringnow quantum fluctuations, we go off-shell and there are two possible AdS-invariantquantizations of the bulk spinor: the conventional one with ψo set to zero and another,‘alternate’ one, with χo set to zero. Whenever the mass of the bulk spinor liesin the window 0 < m < 1

2 , both kinds of modes are normalizable (finite energyconfigurations [27]). Their contribution to the partition function can be computedin the standard way via the Green function for the conventional modes (λ+), andanalytic continuation to account for the alternate modes (λ−). With this choice athand, we can emulate the scalar case [28] since now double quantization for bulkspinors is established. The relative change in the partition function, upon functionalintegration of the quantum fluctuations at quadratic order, is then given by the ratioof the associated functional determinants:

Z+grav

Z−grav

=det+{/∇X

+m}

det−{/∇X+m}

. (5)

3. Boundary double-trace deformation

These two choices of asymptotic behavior correspond in AdS/CFT to two CFT’s [27,29] that share the same field content but differ in the dimension of the fermionicoperator O, dual to the bulk spinor. Despite its appearance, the situation is by nomeans symmetric; the UV CFT with O−, perturbed by the relevant double-tracedeformation O2

−, flows into the IR CFT with O+ (cf. [19, 22, 27]).


3.1. O(1) contribution to the partition function: a shortcut

To get a handle on the relative change in the CFT partition functions at the endpoints of the RG flow, instead of considering the auxiliary field trick as in [19, 22], wesimply adapt the heuristic argument given in [30] to relate ZUV to ZIR. Namely, inthe path integral of the UV CFT we promote the sources η and η to dynamical fieldsand integrate over them

ZIR

ZUV=

∫

DηDη 〈 exp

∫

(ηO− + O−η) 〉 . (6)

The expectation value can be approximated, at leading large N due to the factorizationof the correlation functions, by

exp

∫

η〈O− O−〉η , (7)

and the Gaussian integral results in the functional determinant of the two-pointfunction

ZIR

ZUV= det 〈O− O−〉 , (8)

or, alternatively,

ZUV

ZIR= det 〈O+ O+〉 . (9)

4. The holographic formula

We have now all necessary ingredients to write down the ‘spinor holographic formula’that stems from the postulated equality of the partition functions in AdS/CFTcorrespondence, at subleading order O(1):

det−{/∇X+m}

det+{/∇X+m}

= det 〈Oλ Oλ〉M (10)

The ‘+’ means that we compute with the standard λ in the asymptotic behaviorof the bulk spinor, whereas ‘−’ means the analytic continuation λ → n − λ. Asusual, to make sense out of this formula one needs to tame the divergencies thatarise due to the IR divergent volume of AdS and the UV divergent short distancesingularities. Both divergencies turn out to be tied by the IR/UV connection inAdS/CFT correspondence [31].

This formula conjecturally applies to bulk geometries X which are Euclideansections of asymptotically locally AdS (ALAdS). In particular, when the conformalinfinity M belongs to the conformal class of the standard round spheres, and thereforeconformally flat, the bulk is locally AdS and the IR-divergent volume of AdS factorizes.One can then read off an O(1) contribution to the holographic trace anomaly in evenn, just as in the case of a bulk scalar [3, 7, 32]. Alternatively, from the differenceof one-loop effective actions one can compute the holographic type-A trace anomalycoefficient a following the general recipe of [33]. The behavior of this coefficient for evenn and of a related quantity for odd n, in the cases we will explore, gives support to aconjectured c-theorem valid in all dimensions [21] § and to an F-theorem proposal [22]as well.

§ This promises to settle the disparity pointed out in [34] that, although the computation on theAdS side contemplates even and odd dimensions on equal footing, it is not clear how to translate the“holographic” central charge into field theory language in the case of odd-dimensional CFT’s.


5. The canonical case

As might be expected, the ball model for hyperbolic space (Euclidean AdS) turns outto be the simplest bulk background where calculations can be spelled out in detailand related to the CFT on the conformal boundary (the conformally flat class of thestandard round sphere).

5.1. Bulk

The effective action for a Dirac spinor in hyperbolic space has been recentlyrevisited [15, 16] in connection with a curious gauge-gravity duality where Barnes’multiple gamma function plays a central role. We briefly survey the relevant steps inthe computation of the effective action

S+grav = − log det{/∇+m} . (11)

In terms of the Green’s function, one has (/∇+m)D = −I,

S+grav =

∫ m

trD(n+1) , (12)

where also the spinor indices are traced out. There is a subtlety regarding thedimensionality of the representations of the gamma matrices: for n odd, bulk andboundary representations share the same dimensionality; whereas for n even, in orderto have a Dirac fermion on the boundary, the dimensionality of the bulk representationmust be doubled [27].

We refer to [15, 16] for details of the implementation of dimensional regularizationand the nontrivial role of the bulk volume. In all, one gets a remarkable result, validfor both even and odd dimensions, in terms of Barnes’ multiple gamma

logdet+{/∇+m}

det−{/∇+m}= −21+⌊n

2 ⌋ · logΓn+1(

n+12 +m)

Γn+1(n+12 −m)

. (13)

In addition, whenever n is even one can read off the trace anomaly as in the scalarcase [11, 12]. In the present case one essentially gets the integral of the spinorPlancherel‖ measure(cf. [35]) times the volume anomaly Ln+1 = 2(−π)

n2 /Γ(1 + n

2 )[

2

(2π)n2

∫ m

0

dµ(12 + µ)n

2· (12 − µ)n

2

(12 )n2

]

· Ln+1 . (14)

5.2. Boundary

For the round n-sphere as conformal boundary, the knowledge of the eigenvaluesof the two-point function 〈Oλ Oλ〉Sn and their degeneracies allows a brute-forcecomputation of the corresponding functional determinant. The appropriate basis isthat of spinor spherical harmonics, and the eigenvalues¶ and degeneracies have beenrecently computed in connection with fermionic double-trace deformations [19]

eigenvalues: ±Γ(l + n/2 + ν + 1/2)

Γ(l + n/2− ν + 1/2), (15)

‖ Difference of the effective Lagrangians. The shorthand notation involves Pochhammer’s symbol

(x)n ≡Γ(x+n)Γ(x)

.

¶ The eigenvalues can be also read off from the scattering problem in Hn+1 [36].


degeneracies: 2⌊n2 ⌋ (l + n− 1)!

l! (n− 1)!. (16)

The formal trace is then assembled+ as follows:

log det 〈Oλ Oλ〉Sn = 21+⌊n2 ⌋

∞∑

l=0

(n)ll!

logΓ(l + n+1

2 + ν)

Γ(l + n+12 − ν)

, (17)

and it can be worked out within dimensional regularization as in [11, 22]

log det 〈Oλ Oλ〉Sn = −21+⌊n2 ⌋ Γ(−n)

∫ ν

0

dµ

{

Γ(n+12 + µ)

Γ(1−n2 + µ)

+ (µ → −µ)

}

. (18)

This very same regularized answer we have already found in [16], (eqn.5), whereBarnes’s multiple gamma turned up. The renormalized value can be written, modulopolynomial and logarithmic terms (we refer again to [16] for further details and eqn.6),as the following quotient of Barnes’ multiple gamma functions

log det 〈Oλ Oλ〉Sn = 21+⌊n2 ⌋ log

Γn+1(n+12 + ν)

Γn+1(n+12 − ν)

. (19)

As explained in [16], the gamma factor in front of (18) is deceiving: only for n eventhere is a pole and from the residue one can read off the (integrated) conformal anomalyunder conformal rescaling of the metric

22+n2 (−1)n/2

n!

∫ ν

0

dµ (1

2+ µ)n

2· (1

2− µ)n

2. (20)

At this point we have a perfect match between bulk and boundary computations. Toillustrate the usefulness of this result, besides being an explicit corroboration of theholographic formula, we study a particular value of the spinor mass that unveils theDirac operator on the boundary and connects with a vast mathematical literature ondeterminants of differential operators on spheres (cf. [37]-[43]).

6. Holographic life of Dirac operator

There are two direct ways to identify the Dirac operator at the boundary.

• First, consider the action of the Dirac operator on the flat space two-point function(eqn. 3)

/∇〈O+(~x) O+(~0)〉Rn ∼I

|~x|n+1+2m. (21)

Here one can recognize the Laplacian ∇2 in the limit m → 12 , in a distributional

sense.

• Second, by simple inspection of the eigenvalues of the kernel 〈O+ O+〉Sn on then-sphere (eqn.15) in the same limit m→ 1

2

± (n

2+ l) . (22)

+ In dimensional regularization, the sum over degeneracies of a constant term vanishes, this is whywe do not worry much about the fact that half of the eigenvalues are negative.


6.1. Type-A trace anomaly and Polyakov formulas

We are interested in the universal type-A component of the trace anomaly, thecoefficient of the Euler density or Pfaffian according to the classification in [17]. Forconformally related metrics g = e2wg, in the conformally flat class, Branson (see, e.g.,[37]) conjectured the following Polyakov-like formula

− logdet /∇

2

det /∇2 = c(n)/∇2

∫

M

w(

Qn dvg + Qn dvg

)

+ ..., (23)

where the Pfaffian is traded by Branson’s Q-curvature Qn, a central object inconformal geometry with a much simpler (in fact, linear in w) transformation ruleunder conformal rescalings.

This very same structure we read from the bulk computation, the Q-curvatureterms come from the finite conformal variation of the renomalized volume [44, 45, 46]and the overall coefficient c(n)

/∇2is obtained from the effective Lagrangian at m = 1

2

c(n)/∇2

= 4cn

2

(2π)n2

∫ 12

0

dν(12 + ν)n

2· (12 − ν)n

2

(12 )n2

, (24)

where ck = (−1)k

22kk!(k−1)!. All values reported in [37] for c(n)

/∇2are correctly reproduced by

this formula. Interesting spectral invariants, as zeta function at zero argument andconformal index (cf. [37, 12]), are easily obtained from this coefficient.

6.2. Determinant of iterated Dirac on spheres

The particular value of the determinant of the scattering operator at the mass value12 results in the following remarkable expression in terms of Barnes’ multiple gammafunction, after use of recurrence (A.3),

− log det /∇2 = 4 · 2[n2

]

· log Γn (n

2) . (25)

Barnes’ multiple gamma function is known to occur in functional determinantsof Laplacians on spheres (see, e.g., references in [40]). And yet this compactexpression, which correctly reproduces all zeta-regularized values reported in theliterature(cf. [39, 43]), does not seem to have been noticed until now.

A small digression on a conjecture put forward by Bar and Shopka [39]: theyobserved that the numerical values of these determinants tend to 1 as the dimension ngrows; this was proved in [41] not only for the Dirac operator, but also for the Yamabeor conformal Laplacian on the n-sphere. Our result indicates that the above findingsamount to establishing the limiting value of Barnes’ gamma Γn(n/2) as n → ∞.Furthermore, we can interpret quite naturally this limiting value by looking at thebulk side of the holographic formula: the two boundary conditions coincide as ngrows, or alternatively, λ± → n

2 so that both determinants in the quotient approachone another. This very same argument would predict the same limiting value for allother operators on the right side of similar holographic formulas; this is the case for thedeterminants of GJMS operators on round spheres [12]. In consequence, we predictthe same limiting value of unity; a prediction that can be probed by the asymptoticanalysis of the explicit results obtained in [42] (see eqn.19 therein). In fact, thequotient of Barnes’s gammas in the limit of large dimension d and fixed order k tendsto 1, as was the case for the Dirac operator; at the same time, the reported numerical


values for the multiplicative anomaly M(d, k) hint at a vanishing limit value, so thatthe exponential of both contributions in eqn.(19) of [42] should again render 1 in thelimit.

7. C-theorem proposals and entanglement entropy

Our computations contain and, at times, generalize several scattered results that hadbeen independently derived, most of them in the pursuit of extensions of the C-theoremto higher dimensions and in connection with certain universal terms in entanglemententropy. We briefly list few of them.

Holographic C-charge at O(1):

• n = even, relative change at order O(1) in Cardy’s central charge computedin [19]. CUV − CIR > 0 in the mass window 0 < m ≤ 1

2 . Agreement with theuniversal log-term in entanglement entropy [21].

We find agreement between bulk and boundary outcomes - tables (1) and (2)in [19]- and our eqns. (14) and (20), respectively. Our calculation accounts forthe overall coefficient as well.

• n = odd, relative change at order O(1) in one-loop effective action, candidate forcentral charge, computed in [19]. CUV −CIR > 0 in the mass window 0 < m ≤ 1

2 .Agreement with the universal constant term in entanglement entropy [21].

We find agreement between bulk outcome -integral of eqn.(3.37) in [19]- and oureqn.(13). We are also able to fill the ‘hole’ left in [19] by identifying this finitecontribution on the CFT side, including the overall coefficient as well (eqn.19).

F-coefficient of odd-dimensional CFT’s:

• F-coefficients for free massless Dirac fermions on odd-spheres, table (2) in [22].

This agrees with our result∗ in eqn.(25) and, of course, with the values reportedin [39, 41]. The decrease of the numerical values in the above table is preciselythe hint for the conjecture by Bar and Schopka.

• Under RG flow triggered by a fermionic double-trace deformation, at leadinglarge N, the change in free energy in three dimensions is given by eqn.(82),[22].FUV − FIR > 0 in the mass window 0 < m ≤ 1

2 .

This coincides again with our bulk (eqn.13) and boundary (eqn.19) results.

Entanglement entropy:

• Universal terms in entanglement entropy [21]: logarithmic and constant in evenand odd dimensions, respectively. In the case of a free boson, they coincidewith the holographic anomaly and determinant of Yamabe operator or conformal

∗ The evaluation of the corresponding Barnes’ gammas can be performed with the relations collectedin Appendix A, and one can rewrite in terms of Riemann zeta by use of the relation ζ′(−2n) =(−1)n(2n)!

21+2nπ2n ζ(1 + 2n) .


Laplacian, for even [42] and odd [47] dimensions, respectively.

This matching should hold as well for the entanglement entropy of a free Diracspinor and the trace anomaly and determinant of the Dirac operator on n-spheres.Now under the guise of holographic C-charge at O(1), in the notation of [21], theyverify (a∗d)UV > (a∗d)IR.

8. Conclusion

Our main contribution has been the spinor version of the holographic formula thatconnects the functional determinant of a bulk spinor with that of the two pointfunction of the dual operator at the boundary. The case of pure AdS allows forexplicit computations which, in turn, encompass several results independently derivedin other contexts. In particular, contact is made in the case of the Dirac operatoron the boundary; here we have obtained a compact expression for the determinanton spheres in terms of Barnes’ gamma function as well as the generic type-A traceanomaly in any even dimension.

We have also gained insight into the conjecture by Bar and Schopka and unveiledits possible holographic roots. It seems plausible that the conjecture should alsoapply to the functional determinant of any conformally covariant differential operator,provided a corresponding holographic formula exits. Natural candidates for the bulkside of such holographic formula are the functional determinants in the one-loopeffective action of higher-spin fields (see, e.g., recent progress in [48]).

Further study of the holographic formula, for instance in black-hole backgrounds,remains a challenge. It seems worth to explore the connection of the functionaldeterminants involved in the formula with regularized products of quasinormalfrequencies [9] or scattering resonances. Another possible avenue concerns anintriguing feature of Barnes’ double gamma at integer values, it equals the resultof ordinary determinants of Hankel matrices; this might well be a sign for localizationof the functional integrals that we have been computing in AdS (see, in this respect,[49]).

We close by noticing that the explicit expressions, as obtained via the holographicapproach, come out in a gracious, simple form. This was the case for the trace anomalyof GJMS operators [12] and now for the determinant and trace anomaly of Diracoperator.

Acknowledgments

We thank A. Montecinos for collaboration in the initial stages of this study. This workwas partially funded through fondecyt-chile 11110430, UNAB DI-21-11/R and UNABDI-26-11/R.

Appendix A. Barnes’ multiple gamma: disambiguation

There are several choices for the normalization of Barnes’ multiple gamma functionΓn(z). For definiteness, we stick to the convention in [43, 50, 51] and list few propertiesthat are relevant to our calculations.


• Its logarithm can be written in terms of derivatives of Hurwitz zeta function

log Γn(z) =n−1∑

k=0

bn,k(z) · ζ′(−k, z) , (A.1)

where bn,k(z) is a polynomial in z with Stirling numbers of the first kind s(n, j)in its coefficients

bn,k(z) =(−1)n−1−k

(n− 1)!

n−1∑

j=k

(jk

) · s(n, j + 1) · zj−k . (A.2)

• Recurrence or ladder relation:

Γn+1(1 + z) =Γn+1(z)

Γn(z). (A.3)

• Pascal triangle by successive applications of the ladder relation:

log Γn(m+ z) =

m∑

l=0

(−1)l (ml

) · log Γn−l(z) , 0 ≤ m ≤ n− 1 . (A.4)

• Particular values in terms of derivatives of Riemann zeta function♯:

log Γn(1) =n−1∑

k=0

bn,k(1) · ζ′(−k) , (A.5)

log Γn(1

2) =

n−1∑

k=0

bn,k(1

2) · (2−k − 1) · ζ′(−k)− log 2

n−1∑

k=0

bn,k(1

2)2−kBk+1

k + 1, (A.6)

where Bn are Bernoulli numbers.

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♯ Relation A.5 can be further simplified to log Γn(1) = 1(n−1)!

∑n−1j=1 |s(j, n − 1)| · ζ′(−k), cf. [38]

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